The second statement of Proposition REF  does not necessarily hold if \(G\)  is not a forest. To see this consider the graph \(H\) , which is obtained from \(C_5\)  by adding an open twin to any vertex of the cycle. See Figure REF , where two copies of \(H\)  are depicted. Notice that two different \(\gamma _{gr}^{t}(H)\) -sets are depicted in these two copies of \(H\) , where vertices of a \(\gamma _{gr}^{t}(H)\) -set in each copy are black. It is thus clear that \(H\)  is not an iso-unique Grundy total domination graph. However, \(H-u\)  is isomorphic to \(C_5\) , and it is known [1]} that \(\gamma _{gr}^{t}(C_n)=n-1\)  if \(n\ge 3\)  is odd, and by symmetry of \(C_n\)  it follows that every odd cycle is an iso-unique Grundy total domination graph.
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